High-order harmonics generated by the interaction of intense lasers with atoms or molecules have been extensively studied and provide an effective scheme for achieving attosecond pulses.^[1–8] High-order harmonic generation (HHG) is a highly nonlinear process governed by the dynamic response of atoms in a gaseous medium under the action of a driving laser pulse. The HHG measured in an experiment is the collective response of all atoms in the macroscopic medium. The intensity of the high-order harmonics is sensitive to the driving laser parameters and the properties of the gaseous medium.^[9–14] At present, the application of HHG is limited by a low emission intensity. To increase the emission intensity, one effective method is to generate favorable phase-matching conditions during HHG.

By adjusting the position of the gas target relative to the laser focus as well as the gas density, laser intensity, and other macroscopic parameters, good phase matching can be achieved. The effect of these macroscopic parameters on HHG can be reflected by the long and short quantum trajectories.^[15] Previous studies have generally concluded that the long quantum trajectory is dominant when the gas target is located before the laser focus, while the short quantum trajectory is dominant when the gas target is located after the laser focus. For example, in 2011, Vozzi et al. applied a 1500-nm, 20-fs laser pulse to Xe atoms and observed that when the gas target was located before the laser focus, the long quantum trajectory dominated; moreover, when the gas target was located after the laser focus, the short quantum trajectory dominated.^[16] In 2014, Hutchison et al. applied an 800-nm laser and an orthogonally polarized second-order harmonic field to Ar atoms.^[17] The authors found that when the gas target was in front of the laser focus, the short quantum trajectory on the axis played a greater role, while the long quantum trajectory off the axis was dominant. The harmonic emissions of the long and short quantum trajectories can be separated, and in some cases, intermediate rings generated by the interference of the quantum trajectories can be observed. In the same year, Ye et al. applied an 800-nm, 4.2-fs pulse to Ne atoms and found that when the gas chamber was located before the laser focus, the effect of the long quantum trajectory was prominent; in contrast, when the gas chamber was located after the laser focus, the effect of the short quantum trajectory was prominent.^[15]

However, in our recent work, we found that when the center of the gas target is located at the laser focus, an ultra-short laser pulse can undergo significant reshaping as the gas density increases at moderate laser intensities.^[18] The long quantum trajectory is effective for phase matching at low gas density, while the short quantum trajectory is effective for phase matching at high gas density. Thus, the effects of the long and short quantum trajectories are not simply determined by the position of the gas target relative to the laser focus. The effect of the laser intensity on the quantum trajectory is also an important factor to consider for HHG, in addition to the position of the gas target center relative to the laser focus. The main purpose of this work is to study the effect of the laser intensity on quantum trajectories in the macroscopic HHG process at different gas densities.

In this study, the single-atom response is obtained from the quantitative re-scattering (QRS) model,^[19] and the macroscopic effect is considered by applying Maxwell’s wave equations.^[20] In the simulations, a few-cycle 800-nm laser interacts with Ne atoms. The results indicate that the laser intensity and gas density affect the harmonic emission of the long and short quantum trajectories. For a lower laser intensity, phase matching of the long quantum trajectory dominates at lower and higher gas densities. In contrast, for a higher laser intensity, phase matching of the long quantum trajectory dominates at lower gas densities, while phase matching of the short quantum trajectory can be achieved at higher densities.

In the calculations, the driving laser field parameters are as follows. The wavelength is 800 nm, the pulse duration is 4 fs, and the laser beam waist is 25 μm. The length of the gas target is 1 mm and the gas target center is located 1 mm before the laser focus. An Ne atom is used as the target atom. We show the changes in macroscopic HHG spectra for varying gas densities and four laser intensities in Fig. 1. First, at a low laser intensity of 2 × 10¹⁴ W/cm², as the gas density increases, the efficiencies of the harmonics and the cut-off energies of the harmonic spectra remain almost constant, as shown in Fig. 1(a). As shown in Figs. 1(a)–1(d), the efficiencies of the harmonics increase significantly as the laser intensity increases. At higher laser intensities, as the gas density increases, the efficiencies of the harmonics first increase and then decrease, while the cut-off energies gradually decrease. Clearly, there is an optimal balance between the efficiencies of the harmonics in the plateau and the cut-off energies at a certain gas density; the efficiency of the harmonic is higher and the cut-off energy is larger at this gas density, which is denoted as the optimum gas density. The green dotted lines in Fig. 1 present the HHG spectra at the optimum gas density for each laser intensity.

Fig. 1.

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	Fig. 1. Macroscopic HHG spectra for different gas densities at four laser intensities: (a) 2 × 1014 W/cm2, (b) 4 × 1014 W/cm2, (c) 6 × 1014 W/cm2, and (d) 8 × 1014 W/cm2. The green dotted lines present HHG spectra measured at the optimum gas density for each laser intensity.

To understand the variation in the HHG spectra with laser intensity and gas density shown in Fig. 1, we carefully studied the time-frequency profiles of the near-field harmonics on the end surface of the gas target at lower and higher gas densities for different laser intensities, as shown in Fig. 2. For a laser intensity of 2 × 10¹⁴ W/cm², at both a lower gas density of 1.815 × 10²⁵ atoms/m³ and a higher gas density of 3.134 × 10²⁵ atoms/m³, between 0 and 1.0 optical cycle (o.c.), there is only one primary emission burst per half optical period with a negative chirp, which is generated by the long quantum trajectory, and the corresponding electron ionization occurs between −0.5 and 0.5 o.c. Meanwhile, the positive chirp emission is very weak, indicating that the long quantum trajectory contributes to the emission of the harmonics, as shown in Figs. 2(a) and 2(b). For a laser intensity of 4 × 10¹⁴ W/cm² at both a lower gas density of 1.650 × 10²⁵ atoms/m³ and a higher gas density of 2.969 × 10²⁵ atoms/m³ and for a laser intensity of 6 × 10¹⁴ W/cm² at a lower gas density of 1.485 × 10²⁵ atoms/m³, between 0 and 1.0 o.c., there is one emission burst with two branches per half optical period. The positive chirp emission is generated by the short quantum trajectory and the negative chirp emission is generated by the long quantum trajectory, as shown in Figs. 2(c)–2(e). The corresponding electron ionization occurs between −0.5 and 0.5 o.c. However, the emission intensities of the positive and negative chirps between 0 and 0.5 o.c. are similar, indicating that the short and long quantum trajectories both contribute to the harmonic emission. The negative chirp emission between 0.5 and 1.0 o.c. is significantly greater than the positive chirp emission, implying that the long quantum trajectory is more favorable to the harmonic emission. For a laser intensity of 6 × 10¹⁴ W/cm² at a higher gas density of 2.969 × 10²⁵ atoms/m³, between 0 and 1.0 o.c., we observe one emission burst with two branches per half optical period. However, the negative chirp emission is weak and the positive chirp emission dominates, as shown in Fig. 2(f), indicating that the stronger harmonic emission comes from the short quantum trajectory. As the laser intensity is further increased to 8 × 10¹⁴ W/cm², for both a lower gas density of 1.155 × 10²⁵ atoms/m³ and a higher gas density of 2.474 × 10²⁵ atoms/m³, the negative chirp emission is clearly caused by the long quantum trajectory and the positive chirp caused by the short quantum trajectory is dominant, as shown in Figs. 2(g) and 2(h), respectively. As the laser intensity increases, the effects of the long and short quantum trajectories differ significantly at lower and higher gas densities.

Fig. 2.

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	Fig. 2. Time–frequency profiles of HHG on the end surface of the gas target at lower and higher gas densities for different laser intensities: (a) and (b) 2 × 1014 W/cm2, (c) and (d) 4 × 1014 W/cm2, (e) and (f) 6 × 1014 W/cm2, (g) and (h) 8 × 1014 W/cm2.

Figure 2 only shows information regarding the near-field harmonic emissions on the end surface of the gas target. To understand the mechanism governing the dominant quantum trajectory for a given laser intensity and gas density, we must assess how the harmonic emissions accumulate during propagation in the gas target. In Fig. 3, we show the time–frequency profiles of HHG for selected planes perpendicular to the direction of propagation at different gas densities for two laser intensities. The results for z = −0.5 mm are shown in Fig. 2. Figures 3(a)–3(d) correspond to a laser intensity of 2 × 10¹⁴ W/cm² and a gas density of 1.815 × 10²⁵ atoms/m³, figures 3(e)–3(h) correspond to a laser intensity of 8 × 10¹⁴ W/cm² and a gas density of 1.155 × 10²⁵ atoms/m³, and figures 3(i)–3(l) correspond to a laser intensity of 8 × 10¹⁴ W/cm² and a gas density of 2.474 × 10²⁵ atoms/m³. Figures 3(a), 3(e), and 3(i) show the harmonic emissions at the initial surface of the gas target for three conditions, which all reflect the emission characteristics of a single atom. For a laser intensity of 2 × 10¹⁴ W/cm² and a gas density of 1.815 × 10²⁵ atoms/m³, two primary emission bursts are observed between 0 and 1.0 o.c., and the long quantum trajectory is clearly dominant. As the harmonic field continues to accumulate, the cut-off energy of the harmonics increases significantly. Moreover, the normalization factor indicates that the intensities of the harmonics also increase significantly. However, the relative strengths of the long and short quantum trajectories remain nearly constant, as shown in Figs. 3(a)–3(d). In contrast to the case of low laser intensity, for a high laser intensity of 8 × 10¹⁴ W/cm², two primary emission bursts occur between 0 and 1.0 o.c. regardless of the gas density, indicating that the long and short quantum trajectories exist simultaneously. At a low gas density of 1.155 × 10²⁵ atoms/m³, the harmonic intensities increase with the accumulation of the harmonic fields. The emissions of the long and short quantum trajectories from 0 to 0.5 o.c. are obviously enhanced, while the emission of the short quantum trajectory is weakened from 0.5 to 1.0 o.c., as shown in Fig. 3(f). Figures 3(f)–3(h) show that the short quantum trajectory is weakened near 0 and 0.5 o.c., while the long quantum trajectory dominates near 0.3 and 1.0 o.c. In Figs. 3(h)–3(l), the harmonic emission structure is very stable. At a high gas density of 2.474 × 10²⁵ atoms/m³, the short quantum trajectory between 0 and 0.5 o.c. becomes dominant as the harmonic field gradually accumulates, and the long and short quantum trajectories are equivalent between 0.5 and 1.0 o.c., as shown in Fig. 3(j). The intensities of the long and short quantum trajectories between 0.5 and 1.0 o.c. are almost the same, as shown in Fig. 3(j). In Figs. 3(j)–3(l), the short quantum trajectory is still dominant between 0 and 0.5 o.c.; meanwhile, the long quantum trajectory gradually weakens between 0.5 and 1.0 o.c., and the short quantum trajectory gradually becomes dominant. In Fig. 3(l), the harmonic emission behavior is very stable. Thus, the cut-off energies of the harmonics are significantly reduced during the propagation process.

Fig. 3.

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Fig. 3. Evolution of the time–frequency profiles of harmonic emissions along the direction of laser propagation. The laser intensities and gas densities are as follows: (a)–(d) 2 × 1014 W/cm2, 1.815 × 1025 atoms/m3, (e)–(h) 8 × 1014 W/cm2, 1.155 × 1025 atoms/m3, (i)–(l) 8 × 1014 W/cm2, 2.474 × 1025 atoms/m3. The harmonic efficiency of each position is independently normalized, and the normalization coefficient is given in the lower left corner.

To further elucidate the results presented in Fig. 1, we first performed a detailed analysis for a laser intensity of 8 × 10¹⁴ W/cm² and a gas density of 1.155 × 10²⁵ atoms/m³. Figure 4 shows the evolution of the total phase mismatch |Δk| of the off-axis (r = w/3) HHG along the propagation direction. The blue (red) region corresponds to good (poor) phase matching. The time interval of the black dotted line (red dotted line) frames in Figs. 4(a)–4(d) corresponds to the ionization time of the emission of the long quantum trajectory between 0 and 0.5 o.c. (0.5 and 1.0 o.c.) in Figs. 3(e)–3(h). The time interval of the black dotted line (red dotted line) frames in Figs. 4(e)–4(h) corresponds to the ionization time of the emission of the short quantum trajectory between 0 and 0.5 o.c. (0.5 and 1.0 o.c.) in Figs. 3(e)–3(h). By comparing Fig. 4(a) with Fig. 4(e), it can be seen that at z = −1.25 mm, within the black dotted line frame, the phase mismatch of the harmonics of the long quantum trajectory is greater than that of the short quantum trajectory, resulting in a greater contribution to the harmonics from the short quantum trajectory. The phase mismatches of the harmonics for the electrons emitted by the long and short quantum trajectories in the red dotted line frame are comparable, leading to similar contributions to the harmonics from the long and short quantum trajectories. This finding is consistent with the time–frequency profile of HHG presented in Fig. 3(f). As the harmonic field propagates forward, we find that the phase mismatch of the harmonics of the long quantum trajectory gradually decreases, as shown in Figs. 4(a)–4(d); however, the phase mismatch remains nearly constant for the short quantum trajectory, as shown in Figs. 4(e)–4(h). Thus, the long quantum trajectory becomes dominant during the process of propagation. For a laser intensity of 2 × 10¹⁴ W/cm² and a gas density of 1.815 × 10²⁵ atoms/m³ and for a laser intensity of 8 × 10¹⁴ W/cm² and a gas density of 2.474 × 10²⁵ atoms/m³, the total phase mismatch |Δk| of the off-axis HHG along the propagation direction is shown in Figs. 5 and 6, corresponding to the evolution of the time–frequency profiles of the HHG in Figs. 3(a)–3(d) and 3(i)–3(l), respectively. For a lower laser intensity of 2 × 10¹⁴ W/cm², the phase mismatch of the harmonics of the long quantum trajectory gradually decreases, and thus, the long quantum trajectory is dominant for the HHG. For a laser intensity of 8 × 10¹⁴ W/cm² and a gas density of 2.474 × 10²⁵ atoms/m³, the phase mismatch of the harmonics of the short quantum trajectory remains nearly constant and is always smaller than that of the long quantum trajectory, which results in the dominance of the short quantum trajectory during propagation.

Fig. 4.

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	Fig. 4. Evolution of the total phase mismatch |Δk| of the off-axis HHG along the propagation direction (z = −1.25 mm, −1 mm, −0.75 mm, −0.5 mm): (a)–(d) long quantum trajectory, (e)–(h) short quantum trajectory. The laser intensity is 8 × 1014 W/cm2, and the gas density is 1.155 × 1025 atoms/m3.

Fig. 5.

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	Fig. 5. Evolution of the total phase mismatch |Δk| of the off-axis HHG along the propagation direction (z = −1.25 mm, −1 mm, −0.75 mm, −0.5 mm). The laser intensity is 2 × 1014 W/cm2, and the gas density is 1.815 × 1025 atoms/m3.

Fig. 6.

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	Fig. 6. The results presented here correspond to those presented in Fig. 4 for a gas density of 2.474 × 1025 atoms/m3.

Next, we will further explore the contribution of the individual components of the HHG phase mismatch to the total phase mismatch, namely, Δk_g, Δk_p, Δk_n, and Δk_d, to determine which factor plays a key role in the change of the total phase mismatch, which in turn affects the harmonics. The off-axis electric field of the 800-nm laser beam and the spatial distribution of the ionized electron density are shown in Fig. 7 for three different conditions. The variation in the harmonic cut-off energy shown in Fig. 3 can be well understood from the evolution of the electric field. For a lower laser intensity of 2 × 10¹⁴ W/cm², as the laser field propagates forward along the z direction, the laser electric field and the ionized electron density gradually increase, as shown in Figs. 7(a) and 7(d). This behavior also explains the increase in the harmonic cut-off energy during the evolution of the HHG time–frequency profiles shown in Figs. 3(a)–3(d). However, the ionized electron density is approximately five orders of magnitude smaller than the neutral atom density, which leads to a negligible contribution of Δk_p, and the other three terms are all positive, where Δk_g is the influence of the Gouy phase shift of the Gaussian beam, which is constant for a given system. Because the ionized electron density is very low, the change in the neutral atom density can be ignored. As the harmonic field propagates forward, the contributions of Δk_g and Δk_n remain unchanged, and Δk_d depends on the change in laser intensity. From the evolution of the laser electric field shown in Fig. 7(a), it can be seen that the change in the laser electric field with the change in propagation distance decreases; thus, the contribution of Δk_d and the total phase mismatch each gradually decrease. For a higher laser intensity of 8 × 10¹⁴ W/cm² and a lower gas density of 1.155 × 10²⁵ atoms/m³, the ionized electron density shows little change as the laser field propagates forward, as shown in Fig. 7(e), which causes the contributions of Δk_p and Δk_n to remain nearly constant. However, from the evolution of the field shown in Fig. 7(b), it can be seen that the laser electric field first increases slightly and then decreases slightly, which leads to a gradual change in Δk_d from positive to negative. As the harmonic field propagates, the contributions of Δk_g and Δk_n are positive, the contribution of Δk_p is negative, and Δk_p and Δk_d can compensate for the mismatch caused by Δk_g and Δk_n. Moreover, because the orbital coefficient α_i of the long quantum trajectory is more than 20 times greater than that of the short quantum trajectory, the contribution of the Δk_d term of the short quantum trajectory is very small, which leads to a gradual decrease in the total phase mismatch for the long quantum trajectory. However, the total phase mismatch for the short quantum trajectory remains nearly constant as the harmonic field propagates, which is consistent with the description presented in Fig. 4. At a higher gas density of 2.474 × 10²⁵ atoms/m³, as the laser field propagates forward from z = −1.5 mm to z = −1 mm, the laser electric field and ionized electron density gradually decrease, and from z = −1 mm to z = −0.5 mm, these parameters tend to be stable, as shown in Figs. 7(c) and 7(f). The decrease in the laser electric field directly explains the decrease in the harmonic cut-off energy in the evolution of the HHG time–frequency profiles shown in Figs. 3(i)–3(l). From z = −1.5 mm to z = −1 mm, the density of the ionized electrons is very large. At this point, as z increases, Δk_g and Δk_n compensate for the mismatch caused by Δ k_p and Δk_d. Thus, the phase mismatch for the short quantum trajectory in this process is small. From z = −1 mm to z = −0.5 mm, the laser electric field tends to be stable, and the contribution of Δk_d can be ignored. Therefore, the phase matching remains unchanged, consistent with the description given in Fig. 6.

Fig. 7.

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Fig. 7. Off-axis electric field of the 800-nm laser beam and the spatial distribution of the ionized electron density under three different conditions. (a) and (d) Laser intensity of 2 × 1014 W/cm2 and a gas density of 1.815 × 1025 atoms/m3. (b) and (e) Laser intensity of 8 × 1014 W/cm2 and a gas density of 1.155 × 1025 atoms/m3. (c) and (f) Laser intensity of 8 × 1014 W/cm2 and a gas density of 2.474 × 1025 atoms/m3.

We macroscopically studied the effect of the laser intensity on the quantum trajectory in macroscopic harmonic spectra. By analyzing the time–frequency evolution of the harmonic emissions during propagation, we found that the long quantum trajectory at low laser intensities is dominant at both lower and higher gas densities; in contrast, for high laser intensities, the long quantum trajectory dominates at lower gas density while the short quantum trajectory dominates at higher gas density. By using the phase-mismatch formulation, we evaluated the effect of the laser intensity and gas density on the conditions for generating short or long quantum trajectories. In order to observe the motions of electrons, it is necessary to generate an isolated attosecond pulse in experiment. Through this theoretical calculation, it is possible to select the long or short quantum emission trajectories for the generation of isolated attosecond pulse by optimizing the gas density.